Cutting a mesh in half
I would recommend using the bisect tool (new in Blender 2.69):


Enter edit mode (Tab), then press Space and type in Bisect, or select Bisect from the toolshelf (under Add).
Click and drag, to bisect the object.

Press F6 to bring up the options for the tool, and select clear inner, or clear outer depending on which half you don't want.



If you don't want to delete half, but just want to select it:


Instead of step 3, press H to hide the loop:

Hover over the half you want and press L to select linked vertices under the cursor:

Unhide everything (AltH), leaving you with one half of the object selected: